Which domain sounds better, one starting with "my..." or with "your..."

I agree with Angela ^^^.

For SEO purposes, if that's relevant to you, both are bad. It's better to have any letters/words additional to the keyword after the keyword, in the domain-name, than before it. (This is perhaps less important now than previously, though, with EMD's maybe declining a little in value/relevance anyway.)

Awesome!
Thank you very much for the feedback. I meant YourHairDresser.com, but I think I'll use neither of both.

I actually made a thread about this like 6 months ago in the copywriting forum. And the majority of copywriters voted for "my".

Aside from branding/taglines "your" is more of a command. Like "yourpizzaguys.com" tells people "these are your your pizza guys". Like you are commanding them. On the other hand, "mypizzaguys" is more of a subtle implication. The person reads it and thinks "these are my pizza guys". Like they already made the choice themselves. I believe in copy, "your" is better most of the time, like "these are your benefits", as a general guideline. But it really depends on the specific context imo. Since this context is essentially a choice, you want the person to think "this is my choice". It assumes choice for the person. Because when people make a choice, the word they use in their head is "my".

Now, my name was a bit different than the examples in this post, because "my" also jived better with the overall name I used, but I suggest the OP splits tests a few different names. I did this without a site initially using the name in a headline, and just making different versions. That was ultimately the reason I wound up using "my". It out performed all the others. Not to mention some of the better copywriters also chose that same name.

-Red

But out of the two choices, I'd choose YourHairDresser.com -- notice "your", rather than "you."

Agreed, you confused me with the thread title and then the example in your post, using your and the you. yourhairdresser.com gets my vote